Orden SPIRURIDA Chitwood, 1933.

There are N vintages of capital, each of which embodies a distinct generation of technol- ogy. Denote by j = 1, 2, ..., N the vintage of capital. In the static model, a technology is interpreted as a firm. Technology j combines capital j and labor to produce a single final goods:

[(1 − α)k_jρ+ α(γj`)ρ] 1 ρ

here γj denotes the level of technology embodied in capital j. The assumptions I make are

expressed formally as follows.

Assumption 1. [Capital-labor complementarity] ρ < 0.

Assumption 2. [Labor-saving technological progress] γj increases with j.

The elasticity of substitution between capital and labor is given by 1

1−ρ. The first assump-

tion states that, given a technology (embodied in machines), the elasticity of substitution between capital and labor ranges between 0 and 1; that is, capital and labor are comple- mentary inputs. This assumption is consistent with most empirical estimates (e.g. Antras

(2004);Klump et al.(2007);Herrendorf et al.(2015)). Recently,Oberfield and Raval(2014)) use plant-level data from the Census of Manufactures to estimate an average (plant-level) elasticity of substitution of about 0.5 for 1987. The estimated aggregate elasticity in manu- facturing is 0.71 in 1987 and 0.75 in 2007.

Assumption 2 states that technological progress is labor saving in the sense that new technology embodied in new machines requires less labor input per unit of output. Labor- augmenting technological progress is typically assumed in growth models to be consistent with a balanced growth path (Barro and Sala-iMartin (2004). see Acemoglu (2003) and

Jones (2005) for theoretical justifications).

Let kj denote the supply of capital j and let L denote the supply of labor. In order to

isolate the effect of technology in the static model, all the kj are fixed at 1 and the inelastic

labor supply is also normalized to 1. Labor moves freely among firms. I study the labor allocation problem and investigate the effects of technologies on firm size and labor share. The marginal productivity of labor in firm j is

MPLj =  (1 − α) kj ` ρ + αγ_jρ 1/ρ−1 αγ_jρ.

When the employment in firm j approaches zero, the firm’s marginal productivity of labor is MPLj(0) = α1/ργj.

Since capital is fixed at a positive number, it follows that the marginal productivity of labor at zero employment is not equal to infinity (i.e. the Inada condition does not hold). Hence some firms might not hire any labor in equilibrium. In addition, firms that use more advanced technologies have a higher marginal productivity of labor at zero employment. The employment flow always begins with firm j = N and moves downwards, step by step. Thus the first unit of labor goes to the most productive firm, N. As firm N accumulates labor, its marginal productivity of labor declines. As soon as that level falls to the second advanced firm’s marginal productivity of labor (at zero employment), that second firm begins to hire labor. This process continues until full employment is reached.

The labor market equilibrium condition is37 w = MPLj =  (1 − α)  kj γj`j ρ + α 1/ρ−1 αγj.

As a result, technologies that are more advanced (i.e., with a higher γj) will cause a

decline in the adjusted capital/labor ratio, kj/γj`j.38 The labor share in firm j is

LSj = w`j yj = 1 − α α  kj γj`j ρ + 1 −1 .

Because the labor share is an increasing function of kj/γj`j, firms that use more advanced

technologies have a lower labor share.

Technology also affects firm size, which is (as in the concentration data) equal to value added divided by revenue. Combining the formula for firm size and the labor market–clearing condition now yields the following expression:

yj = kj

 1 − α(w/αγj)−ρ/(1−ρ)

1 − α

−1/ρ .

According to this formula, output is greater if the value of γj is higher. In other words, a

more advanced technology increases firm size. Therefore, the single parameter of technology (γj) is enough to generate a negative correlation—as observed in the data—between firm

size and labor share. Formally, the following proposition holds.

37_{This condition holds only for firms that have positive employment in equilibrium.}

38_{I remark that the true capital labor ratio, k}

j/`j, can be either increasing or decreasing in γj. The

direct effect of technologies with a higher γ is to substitute raw labor. Yet that technology also increases the

productivity of capital, which in turn increases labor demand. The equilibrium kj/`j depends on which effect

dominates (see the appendix for a detailed discussion). In Section 3.2, I show that, if capital is adjustable, then the capital/labor ratio is a positive function of the technology parameter, γ.

Proposition 1. [Effects of technology on firm size and labor share]

• If j > j0_{, then LS(j) < LS(j}0_{); that is, firms that use more advanced technologies have}

a lower labor share.

• If j > j0_{, then y(j) > y(j}0_{); that is, firms that use more advanced technologies produce}

more output.

To see the intuition behind this result, recall that technology (γ) both complements capital and increases the productivity of capital, which further increases demand for effective labor (γ`). Hence firms produce more when they use more advanced technologies; however, technology is a substitute for raw labor and so reduces labor’s share of income.